Final answer:
To solve (10 - 4i) ÷ (5 + i), we multiply by the conjugate of the denominator. The result is (54 - 30i)/26, which simplifies to Twenty-three thirteenths minus fifteen thirteenths i.
Step-by-step explanation:
To evaluate the complex number division (10 − 4i) ÷ (5 + i), we should multiply the numerator and denominator by the conjugate of the denominator to remove the imaginary part from the denominator. The conjugate of (5 + i) is (5 - i). So we do the following:
(10 - 4i) * (5 - i)
---------------------
(5 + i) * (5 - i)
Now let's multiply the complex numbers:
(10 * 5 + 10 * -i - 4i * 5 + 4i * i)
-----------------------------------
(5^2 - i^2)
When we simplify:
(50 - 10i - 20i - 4(-1))
-----------------------
(25 + 1)
This simplifies further to:
(54 - 30i)
------------
26
Finally, we divide each term by the denominator:
54/26 - 30i/26
The result is reduced to:
Twenty-three thirteenths minus fifteen thirteenths i.
The correct answer among the options given is therefore Twenty-three thirteenths minus fifteen thirteenths i.